Question

x,y是正数,求证:(x+y)(x^2+y^2)(x^3+y^3)大于等于8x^3y^3.

Answer 1

x+y≥2√(xy)

x²+y²≥2√(x²y²)

x³+y³≥2√(x³y³)

三式相乘,因为都是当x=y时取等号,故得到的不等式也能取等号

(x+y)(x²+y²)(x³+y³)≥8√(x^6y^6)=8x³y³